IN A SMALL BOWL, STIR TOGETHER MUSTARD, MAYONNAISE, BARBECUE SAUCE, AND HOT SAUCE. STORE, COVERED, IN REFRIGERATOR UP TO 3 DAYS.
IN A LARGE BOWL, COMBINE GROUND BEEF, WORCESTERSHIRE, SALT, AND PEPPER. SHAPE GROUND BEEF INTO 6 OVAL PATTIES.
IN A MEDIUM CAST-IRON SKILLET, MELT 2 TABLESPOONS BUTTER OVER MEDIUM HEAT. ADD ONION. COOK, STIRRING OCCASIONALLY, UNTIL ONION IS SOFT AND GOLDEN BROWN, APPROXIMATELY 35 MINUTES.
IN A LARGE CAST-IRON SKILLET, COOK PATTIES OVER MEDIUM-HIGH HEAT UNTIL BROWNED AND COOKED THROUGH, APPROXIMATELY 2 MINUTES PER SIDE.
REMOVE SKILLET FROM HEAT. REMOVE PATTIES FROM SKILLET; WIPE OUT SKILLET.
LAYER 1 BREAD SLICE WITH 1 TABLESPOON SECRET SAUCE, 2 TO 3 TABLESPOONS CARAMELIZED ONIONS, 1 SLICE CHEESE, 1 PATTY, AND ANOTHER 1 TABLESPOON SECRET SAUCE. TOP WITH ANOTHER BREAD SLICE. REPEAT WITH REMAINING BREAD, SECRET SAUCE, CARAMELIZED ONIONS, CHEESE, AND PATTIES.
HEAT SKILLET OVER MEDIUM-HIGH HEAT. MELT 2 TABLESPOONS BUTTER IN SKILLET. WORKING IN BATCHES, COOK SANDWICHES, FLIPPING ONCE, UNTIL GOLDEN BROWN AND HEATED THROUGH, APPROXIMATELY 3 MINUTES PER SIDE. ADD REMAINING BUTTER TO SKILLET AS NEEDED.
